At first I thought I just needed to lubricate my chain, so I degreased it and just re-lubed it very thoroughly. It's still squeaking. I thought it could maybe be a derailleur pulley issue, but I noticed that the squeak only occurs when the chain is pulled forward and not when it freewheels backwards. Is there any condition that would cause a chain/pulley/whatever to squeak in only a single direction?

I understand that torque load can sometimes be a factor, so I tested it by pedaling with minimal (near-zero) force and it still squeaks in only the forward direction, so it doesn't seem to be affected by heavy load vs. light load.

The only change I have made recently was that I took out and regreased my derailleur pulleys, but they were fine for about a week until today when it was lightly raining in the morning. It seems plausible that the rain somehow messed up my pulley greasing, but I don't understand why it would be a unidirectional problem.

If it's squeaking and it's coming from the drivetrain, it is probably the chain. What did you use as lube on the chain and the pulley bushings? Often cleaning a chain (and working degreaser into the links) will make a chain squeak more than before.
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WTHarperFeb 5 '13 at 3:07

How many miles (about) on the chain and sprockets? Can you get any hint as to where the squeak is coming from?
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Daniel R HicksFeb 5 '13 at 3:15

And does it squeak at any particular point in the pedal rotation?
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Daniel R HicksFeb 5 '13 at 3:16

It squeaks consistently throughout the rotation. Yes, I thought that regreasing it could have been a problem, but I only regreased the chain after it started squeaking. It definitely sounds like it is chain/cogset and not coming from the pulleys, though I guess it's possible I'm wrong. The sprocket and chain were replaced at the same time, about a year ago, and I've done about 3000 miles since then. It squeaks regardless of what gear I'm in (though some appear to be slightly louder than others).
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DaralFeb 5 '13 at 4:40

2 Answers
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Since you have 3000 miles on the current chain, it is quite possible that you are nearing the end of the life span of the chain. An easy way to check this is to grab a ruler and measure 12 links. Remember that a full link has both an inner and outer section so a full link will look something like this : (0)=(0)=()
So count every other pivot. Each link should be ~1 inch. 12 links should = 12 inches. If 12 links on your chain is more than 12 1/4 inches you should replace the chain. This would be a good time to check the rest of your drive train for wear too.

From your question, it's uncertain if you are performing the tests with the bike stopped (pedaling backwards) or moving (coasting).

If the squeaking happens when you coast, it could be something in the freewheel, or due to the relative movement between the cogset and the wheel, but I have the impression this is not the case.

Now if the noise appears when you pedal backwards (but not when you coast), and you re-checked pulleys and drivetrain lubing, it could be because your derailer have lateral play, and it changes its angle depending on which direction you spin the pedals.

Just for the sake of completeness, there are some hubs (American Classic, for example), where the freewheel is attached directly to the axle, while usually the cassette is attached to the wheel, which in turn is attached to the axle. I'm saying that because this would be the only situation where it would make a difference if you coast or if you pedal backwards, since there would be no relative movement between cogset and axle while coasting, only while pedaling backwards. In the regular, more common situation, it wouldn't matter with respect to freewheel, only with respect to chain movement.

Of course it's hard to deduce what the problem is only by this description (it is in the flesh, already), but I hope this helps, and please add feedback when you figure it out!

Sorry for the incompleteness. I was testing it by pedaling it around. When coasting it makes no sound, it only squeaks pedaling. That was what I meant when I said "the chain is pulled forward"
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DaralFeb 6 '13 at 1:58